Chance to land a 24 on a game?

There is a game on Neopets, which is called Bilge Dice. My question is: what are the odds to land a perfect score of 24? If there is another variant of this game in real life, I'm sure a few will know.

Basic rules: You have 4 throws to get 4, 1, 6, 6, 6, 6 (for a perfect score). In every try you HAVE to get the 4 and 1, otherwise your throw will not count. I am not really good at explaining things so here is the explanation of neopets.com:

I need the odds for a project I am working on, so any help will um.. yeah.. help. :P

Okay, this thing is harder than I first thought. You can use strategies to improve your chances of winning, and I can't ask you guys to work all that out. It's just too complicated and maybe not even calculatable.

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Bilge Dice Bilge Dice is the latest and greatest dice game to hit Neopia! It takes a little bit of luck and a little bit of strategy to really master. Players take turns rolling a series of dice and the one with the highest total in the end is the winner. There is one catch - players must have secured a roll of 1 and a roll of 4 before the end in order for the player's score to count.

Game Overview
Players begin by rolling six dice. Each turn, you must select one or more dice to remove and be placed in the scoring pool. Once you have kept at least one die, the choice of re-rolling the remaining dice becomes available. If you re-roll the dice, you must keep at least one die before being able to re-roll again. In essence, you choose one or more dice to keep each round until no dice remain. The Strategy
In order for your final score to count, you must also fill the qualification requirements. These can be met by selecting to keep a 1 and a 4 at some point during the game. As you'll see, the first 1 and 4 you select to keep automatically go toward the qualifiers. How the winner is determined
If the player goes through all the dice, and is unable to keep a 1 and 4 before the end, that player is disqualified and loses.
If the player does fill the 1 and 4 requirements, the remaining four dice are added together to form that player's score. Which ever player has the highest score is the winner. In the event of a tie, it is called a "push" and the ante will be returned to the players. Good Luck!